#### Unsteady-State Energy Balances on Tanks: Screencasts

When the valve on an adiabatic tank is opened and some of the ideal gas is released, the temperature and the amount of the remaining gas is calculated as a function of the final pressure in the tank.

We suggest that after watching this screencast, you list the important points as a way to increase retention.

Uses an interactive simulation to describe the impact of the state of the feed to a distillation column on the liquid and vapor flow rates in the column.

We suggest that after watching this screencast, you list the important points as a way to increase retention.

##### Important Equations:

For an ideal gas:

\[\Delta U = C_V\Delta T; \hspace{3mm} \Delta H = C_P\Delta T\]

where \(\Delta U\) is the change in internal energy per mole

\(C_V\) is the constant volume molar heat capacity

\(C_P\) is the constant pressure molar heat capacity

\(\Delta H\) is the change in enthapy per mole

Adiabatic reversible expansion for an ideal gas is applied to the expanding gas that remains in the tank:

\[\frac{T_2}{T_1} = \left( \frac{P_2}{P_1} \right) ^\frac{R}{C_P}\]

where \(P_1\) and \(T_1\) are the initial pressure and temperature

\(P_2\) and \(T_2\) are the final pressure and temperature. The temperatures must be absolute (K), and \(R\) is the ideal gas constant.

Ideal gas law for opening tank valve to release some gas and then closing valve:

\[P_1V = n_1RT_1 \hspace{5mm} P_2V = n_2RT_2\]

where \(V\) is the tank volume

\(n_1\) is the initial number of moles in the tank

\(n_2\) is the final number of moles in the tank

Energy balance for adiabatic evaporation of a fraction of liquid into a vacuum:

\[\Delta U^t = 0\]

where \(\Delta U^t\) is the total energy change of the system (liquid and vapor). The container is assumed to have no heat capacity

\[m_f ^{vap} U_f ^{vap} + m_f ^{liq} U_f ^{liq} = m_i U_i ^{liq}\]

\[m_i = m_f ^{vap} + m_f ^{liq}\]

where \(m_i\) is the initial mass of liquid

\(m_f ^{vap}\) is the final mass of vapor

\(m_f ^{liq}\) is the final mass of liquid

\(U_f ^{vap}\) is the internal energy (per g or per mol) of the vapor at equilibrium

\(U_f ^{liq}\) is the internal energy (per g or per mol) of the liquid at equilibrium

\(U_i ^{liq}\) is the internal energy (per g or per mole) of the initial liquid

\[U^{liq} = U_{ref} ^{liq} + C_V ^{liq} (T – T_{ref})\]

where \(U^{liq}\) is the internal energy of liquid at temperature \(T\)

\(U_{ref} ^{liq}\) is the internal energy of liquid at the reference temperature \(T_{ref}\)

\(C_V ^{liq}\) is the constant-volume heat capacity of the liquid, which is essentially the same as \(C_P ^{liq}\).

\[U^{vap} = U_{ref} ^{liq} + \Delta U_{ref} ^{vap} + C_V ^{vap} (T – T_{ref})\]

where \(U^{vap}\) is the internal energy of the vapor at temperature \(T\)

\(\Delta U_{ref} ^{vap}\) is the change in internal energy for vaporization at \(T_{ref}\)

\(C_V ^{vap}\) is the constant-volume heat capacity of the vapor

At equilibrium, \(T_{liquid} = T_{vapor}\)

Gas phase is assumed to be an ideal gas.

Antoine equation:

\[lnP^{sat} = A – \frac{B}{T+C}\]

where \(P^{sat}\) is the saturation pressure at temperature \(T\)

\(A\), \(B\), and \(C\) are constants